Invariant Gibbs measures and global strong solutions for nonlinear Schr\"odinger equations in dimension two

We consider the defocusing nonlinear Schr"odinger equation on ${\mathbb{T}}^2$ with Wick ordered power nonlinearity, and prove almost sure global well-posedness with respect to the associated Gibbs measure. The heart of the matter is the uniqueness of the solution as limit of solutions to canonically truncated systems. The invariance of the Gibbs measure under the global dynamics follows as a consequence. The proof relies on the novel idea of random averaging operators.